Why do I find math so difficult

Why is math so difficult to learn?

Fear of math sometimes resembles fear of God.
Jaromir Konecny

Many students think that math problems are all about the result; In other words, the pupils are primarily concerned with the recipe for quickly arriving at a certain, namely the right, result. However, this fundamental attitude when learning mathematics is a mistake that often leads to problems later, because for many pupils the essential control mechanism of understanding is switched off by the existing formulas, i.e. In other words, the students orientate themselves only on the way to the result, not on the process. Therefore, one of the most important foundations when learning mathematics is to learn to estimate and to develop a feeling for the plausibility of a calculation process. In this context, one could also say that in every calculation process in mathematics, the most important thing is the meaning of a calculation, not the numbers or formulas alone.

By the way, studies have shown: If you base the learning of mathematics on general, overarching mathematical principles - for example by Add and Multiply does not learn one after the other but at the same time - it is much easier for pupils to understand mathematics.

It is well known that many students complain that mathematics is a so difficult subject to study is. Most adults also agree, looking back, that math was their hardest subject in school, and that they still struggle to understand math problems their own children do today. Even a look at the maths notebooks of schoolchildren can shudder, because it is teeming with cryptic arithmetic and mathematical signs, formulas and abbreviations, unknowns such as x and y, Greek letters and fractions, brackets in all their variations. With a shudder, some still remember that their math teacher was enthusiastic about elegant blackboard letters, but that, as students, they only ever understood the train station.

For example, in a relevant advertising broadcast for tutoring under the title “Mathematics - the subject that most students dread”: “In fact, there is hardly any other area where more tutoring is needed and given than here. Those who have a good command of mathematics can literally earn a golden nose as a mathematics teacher. (...) To this day it is not clear why so many students have problems with this subject. (...) But the same principle applies to mathematics as to languages ​​or any other subject: Tasks and rules have to be repeated a hundred times so that they can be embedded and you can calculate them in your sleep. Nevertheless, particular caution is required with math problems: Just because a problem looks similar to one that you already know and can do does not mean that it will be calculated in exactly the same way. "

In the written and also linguistic notation of mathematics is one of the main causes of the problems people have with math. As logical, comprehensible and convincing as some of the invoices on the blackboard or in your own exercise book at school were, it becomes difficult when you want to repeat or practice the same tasks at home. In a sense, the notation in mathematics only represents the skeleton of every mathematical calculation, while the tendons, the cartilage and the flesh that is located between the bones are missing on the paper. Due to the lack of these connections, there is also no possibility of recalling what has already been understood.


Note: The mathematical symbol language can be frightening and, above all, frightening, because to outsiders it may even sound unnecessarily complicated, as if one wanted to make oneself important with unknown symbols or to pretend complexity in statements that could be represented much more easily in normal words. In truth, the opposite is true, because the language of mathematics may only look confusing if you don't master it, but once you learn it you will find that it makes things clearer and clearer. In the case of mathematical symbols, there is no need to discuss how they are meant, because the scope for interpretation is minimal compared to normal language. So the language of mathematics is actually the exact opposite of the language that is commonly used in everyday life, because this is sometimes rather vague, especially if you want to avoid being clearly defined. This essential issue should be discussed in detail at the beginning of every mathematics lesson in order to show the pupils the meaning of the symbolic language.


In addition, there is a lot in mathematics, and much more so than in any other subject, it is clear what is right and what is wrong, d. That is, there is a very sharp limit to distinguishing right and wrong in a calculation. Compared to other subjects, this would actually be an advantage, because in German or other learning subjects, for example, it is not always so clear and comprehensible which answer or which formulation is correct and which is incorrect. The problem is, however, that from the pupil's point of view, this makes the teacher, whether he / she wants it or not, the master of right or wrong, and that establishes power and creates power Fear, and fear destroys motivation in the long run. Right and wrong cannot be argued in mathematics - especially in school mathematics!

To the Children and adolescents enjoy mathematics In the opinion of the mathematician Armin P. Barth, one should always ask challenging but solvable problems at the beginning and never theories whose relevance they can only see later. You have to pick up the students where they are, with tasks that interest them and that have something to do with their everyday lives. To strengthen motivation, you have to consider its three aspects: Your own Experience competence to make possible, one should always encourage and design the lessons in such a way that a sense of achievement is possible. The social acceptance mathematics is reinforced by taking up pupils' suggestions. The third aspect that autonomy is more difficult to achieve, so in math lessons you should always make it clear why it is worth learning certain things and what problems you can overcome with them.

Studies aiming to obtain a holistic picture of mathematics teaching and to establish a connection between certain teaching characteristics and the learning success of pupils have found that learning processes in mathematics are evidently particularly successful when teachers are involved take into account the previous knowledge of your students and systematically link learning content. Since so much depends on previous knowledge, especially in mathematics, it is definitely important to systematically build up specialist skills and to always know exactly what level the students have achieved.

According to the studies, the need for development in the area of cognitive activation to be, because this teaching feature particularly promotes technical understanding and learning success. Cognitive activation is rated as high when the lessons are geared towards understanding and reasoning and the students are confronted with challenging content that is linked to their previous knowledge and experience. In addition, teachers have to learn from their students Give feedback, because they would need more informative feedback that goes into why a calculation method is suitable or unsuitable to support learning progress.

That is also important Class leadershipthat with the Attitude of the teachers This was better when the teachers themselves stated that they enjoyed their work with the respective class. However, what is the cause and what is the effect must remain open here. Finally, the studies have also shown that the more demanding the mathematics subject matter becomes in the course of the lesson, the less interest the students will be interested in. This shows that it is necessary to bring cognitive promotion and motivation together.

The solution to this "math problem"

Students should therefore try in math lessons to note down those spoken connections of the skeleton in their math notebooks in a suitable way so that they will later be able to remember what happens “between the characters” and “between the lines” to remind you. One will often wonder how long a short line of a formula actually is on paper in spoken words.

Incidentally, there is a similar notation problem in the natural sciences such as chemistry or physics, which in some people trigger similar fears and thus often rejection.

By the way: Although the Greeks were not the first to deal with mathematics, the Babylonians and Egyptians did so before them, but it originated Word math from the Greek, where the Greek word mathema as much as what has been learned, knowledge or, in general, science means. The term mathematike techne thus denotes the art of learning or belonging to learning. There is no generally accepted definition of mathematics, but mathematics is usually understood as the science that uses logic to examine abstract structures that it has created itself through logical definitions for their properties and patterns.

Fingers help with learning math

Research has shown, by the way, that a visual aid like your own fingers is a Key function when it comes to understanding and teaching math. People have an image of their fingers in their brains, even if you don't use your hands for arithmetic, which also applies if you are long out of the age to count things with your fingers.

A study with students between the ages of 8 and 13 who were asked to solve complex negative problems showed that the area of ​​the brain for perceiving the fingers was activated, even if the students did not use their hands at all. Your own fingers are probably your best visual aid and are crucial to understanding math and developing your brain, well into adulthood. Understanding mathematics with the help of one's own fingers is probably so crucial that one suspects this is a reason for the often higher mathematical understanding of piano players and other musicians.

Scientists therefore believe that if you prevent children from calculating with their fingers, it hinders their mathematical development. It is well known that many children do not dare to count with their fingers and only do so secretly under the table. In the process, all of the math changes for students who learn through visual representations and thereby gain a new, deeper understanding.

Dupont-Boime & Thevenot (2018) found that children between the ages of five and six, who have particularly good working memories compared to their peers, are more likely to count arithmetic problems on their fingers and achieve good results with this strategy. In an experiment, they had simple addition problems solved with sums below and above ten, whereby a hidden camera was used to record whether the children were using their fingers to help them. In addition, the children's working memory was checked, i.e. the ability to store information for a short time and manipulate it mentally. The participants heard a sequence of numbers that they should remember and then recite in reverse order. These studies found a connection between finger arithmetic, working memory capacity and the number of correct solutions that the children achieved in detail. In addition, children who had good working memories and who did better with tasks also tended to use a more effective strategy for counting math results on their fingers. In contrast to those who laboriously tried to represent both summands with their hands, they only counted upwards from the first summand with their fingers, so they started to count upwards from seven onwards with four fingers, for example in the 4 + 7 task. Dupont-Boime and Thevenot therefore suggest that children with weaker working memories may not be able to figure out this strategy on their own, so children who have difficulty doing arithmetic should be encouraged to use finger arithmetic. On the basis of other studies, it is also assumed that this strategy only makes sense up to an age of about eight years, because afterwards the relationship between arithmetic and finger arithmetic is reversed.

Basic non-numerical skills must be encouraged early on

So that children don't have math problems later, the kindergarten a age-appropriate cognitive development are promoted, whereby in addition to mathematical skills, the development of non-numerical basic skills is important. These include basic skills such as memory, acoustic and visual perception, the ability to concentrate and spatial orientation. Studies show that the non-numerical basic skills are closely related to the basic mathematical skills, because the isolated practice of mathematical skills is not very helpful if you don't do spatial orientation exercises at the same time. If you let children deal with quantities, shapes and numbers at an early age, it will not be difficult for them to develop an understanding of logic, geometry and abstract, mathematical thinking.

Lehrl et al. (2019) conducted a study of the effects of growing up children in a family in which they are encouraged to learn at an early age. It was examined in detail how important the family learning environment is in the early years of life for the development of skills up to puberty. It showed that parents who encourage their preschool children to develop written, linguistic and mathematical skills, for example by playing dice together or looking at picture books, show better reading and math skills in secondary schools. This also applies to support in kindergartens, because earlier studies had shown that educators have a positive influence on children and their mathematical and linguistic development when they work with them picture books read, linguistically accompany everyday situations or Dice and board games play.


In an interview in the Badische Zeitung on June 30, 2018, Albrecht Beutelspacher, head of the “Mathematikum” museum, explains why mathematics seems so complicated to many people. In his opinion, this is very abstract for many and far from what is known. The fact that many people have problems with mathematics is due to two things: “On the one hand there is the mathematical language: Fractional lines, brackets, plus signs ... It's like learning a new language with completely different rules. On the other hand, mathematics has a lot to do with thinking and imagining, so you have to get involved. "

In order for math to be more fun, you should realize that you can figure something out by your own thinking. So you have to want to understand something and be prepared to think a little bit, because you don't get solutions by trying, but have to come up with the right idea at some point. The mathematical terms are important here, i. In other words, it makes sense to learn them and also to talk about them with others, to think about them together and to develop ideas. “It also helps to look around outside: circles, lines, right angles, parallels - our environment is full of mathematics. Once you have that in your head, you can see more. "

See also
The language of mathematics

Didactics in mathematics


By the way, 2018 will be a Issue in the journal for mathematics didactics appear that deals with psychology as the reference discipline of Mathematics didactics employed. There it says: “In numerous studies in the recent past, psychological theories and models have also been used to investigate questions of didactic mathematics. For example, for a better understanding of learning difficulties in the case of breaks, in addition to technical aspects, the cognitive processing processes are also considered. Precisely because there are close relationships between psychology and mathematics didactics, the question arises of specific role of psychology in mathematics didactic research. In which areas of current mathematics didactic research are psychological approaches particularly influential? To what extent are such approaches helpful and useful for answering specific questions relating to mathematics didactics, and what are the limits?
By the way: The selected articles will be invited by the editors in July 2016. The manuscripts must then be completed by December 31, 2016. The magazine will be published in 2018.


literature

Dupont-Boime, J. & Thevenot, C. (2018). High working memory capacity favors the use of finger counting in six-year-old children. Journal of Cognitive Psychology, 30, 35-42.
Lehrl, Simone, Ebert, Susanne, Blaurock, Sabine, Rossbach, Hans-Günther, Weinert, Sabine (2019). Long-term and domain-specific relations between the early years home learning environment and students ’academic outcomes in secondary school. School Effectiveness and School Improvement, doi: 10.1080 / 09243453.2019.1618346.
http://www.spektrum.de/news/kinder-mit-gutem-gedaechtnis-rechner-mit-den-fingern/1553228 (18-03-20)
http://www.badische-zeitung.de/neues-fuer-kinder/wie-eine-neue-sprache–154077608.html (18-06-30)
The mathematician Armin P. Barth in an interview with the Badener Tagblatt on September 25, 2018.
https://www.spektrum.de/kolumne/symbole-in-zeiten-der-pandemie/1722960 (20-04-22)